1. Ninth Synthesis Imaging Summer School Socorro, June 15-22, 2004 Wide Field Imaging I: Non-Coplanar Arrays Rick Perley

2. Introduction From the first lecture, we have a general relation between the complex visibility V(u,v,w), and the sky intensity I (l,m): where This equation is valid for: spatially incoherent radiation from the far field, phase-tracking interferometer narrow bandwidth What is ‘narrow bandwidth’?

3. Review: Coordinate Frame The unit direction vector s is defined by its projections on the (u,v,w) axes. These components are called the Direction Cosines, (l,m,n) The baseline vector b is specified by its coordinates (u,v,w) (measured in wavelengths). u v w s    l m b n

4. When approximations fail us … Under certain conditions, this integral relation can be reduced to a 2-dimensional Fourier transform. This occurs when one of two conditions are met: 1.All the measures of the visibility are taken on a plane, or 2.The field of view is ‘sufficiently small’, given by:  ant ABCD 6 cm9’6’10’17’31’ 20 cm30’10’18’32’56’ 90 cm135’21’37’66’118’ 400 cm600’45’80’142’253’ Table showing the VLA’s distortion free imaging range (green), marginal zone (yellow), and danger zone (red)

5. Not a 3-D F.T. – but let’s do it anyway … If your source, or your field of view, is larger than the ‘distortion- free’ imaging diameter, then the 2-d approximation employed in routine imagine are not valid, and you will get a crappy image. In this case, we must return to the general integral relation between the image intensity and the measured visibilities. The general relationship is not a Fourier transform. It thus doesn’t have an immediate inversion. But, we can consider the 3-D Fourier transform of V(u,v,w), giving a 3-D ‘image volume’ F(l,m,n), and try relate this to the desired intensity, I (l,m). The mathematical details are straightforward, but tedious, and are given in detail on pp 384-385 in the White Book.

6. The 3-D Image Volume We find that: where F(l,m,n) is related to the desired intensity, I (l,m), by: This relation looks daunting, but in fact has a lovely geometric interpretation.

7. Interpretation The modified visibility V 0 (u,v,w) is simply the observed visibility with no ‘fringe tracking’. It’s what we would measure if the fringes were held fixed, and the sky moves through them. The bottom equation states that the image volume is everywhere empty (F(l,m,n)=0), except on a spherical surface of unit radius where The correct sky image, I (l,m)/n, is the value of F(l,m,n) on this unit surface Note: The image volume is not a physical space. It is a mathematical construct.

8. Benefits of a 3-D Fourier Relation The identification of a 3-D Fourier relation means that all the relationships and theorems mentioned for 2-d imaging in earlier lectures carry over directly. These include: –Effects of finite sampling of V(u,v,w). –Effects of maximum and minimum baselines. –The ‘dirty beam’ (now a ‘beam ball’), sidelobes, etc. –Deconvolution, ‘clean beams’, self-calibration. All these are, in principle, carried over unchanged, with the addition of the third dimension. But the real world makes this straightforward approach unattractive (but not impossible).

9. Coordinates Where on the unit sphere are sources found? where  0 = the reference declination, and  = the offset from the reference right ascension.  However, where the sources appear on a 2-d plane is a  different matter.

10. Illustrative Example – a slice through the m = 0 plane Upper Left: True Image. Upper right: Dirty Image. Lower Left: After deconvolution. Lower right: After projection 1 To phase center 4 sources 2-d ‘flat’ map Dirty ‘beam ball’ and sidelobes

11. Snapshots in 3D Imaging A snapshot VLA observation, seen in ‘3D’, creates ‘line beams’ (orange lines), which uniquely project the sources (red bars) to the image plane (blue). Except for the tangent point, the apparent locations of the sources move in time.

12. Apparent Source Movement As seen from the sky, the plane containing the VLA rotates through the day. This causes the ‘line-beams’ associated with the snapshot images to rotate. The apparent source position in a 2-D image thus rotates, following a conic section. The loci of the path is: where Z = the zenith distance, and  P  = parallactic angle, And (l,m) are the correct angular coordinates of the source.

13. Wandering Sources The apparent source motion is a function of zenith distance and parallactic angle, given by: where H = hour angle  = declination  = antenna latitude

14. And around they go … On the 2-d (tangent) image plane, source positions follow conic sections. The plots show the loci for declinations 90, 70, 50, 30, 10, -10, -30, and -40. Each dot represents the location at integer HA. The path is a circle at declination 90. The only observation with no error is at HA=0,  =34. The error scales quadratically with source offset from the phase center.

15. Schematic Example Imagine a 24-hour observation of the north pole. The `simple’ 2-d output map will look something like this. The red circles represent the apparent source structures. Each doubling of distance from the phase center quadruples the extent of the distorted image. l m.  = 90

16. How bad is it? In practical terms … The offset is (1 - cos  ) tan Z ~ (  2 tan Z)/2 radians For a source at the antenna beam half-power,  ~  /2D So the offset, , measured in synthesized beamwidths, ( /B) at the half-power of the antenna beam can be written as For the VLA’s A-configuration, this offset error, at the antenna FWHM, can be written:  ~ cm (tan Z)/20 (in beamwidths) This is very significant at meter wavelengths, and at high zenith angles (low elevations). B = maximum baseline D = antenna diameter Z = zenith distance = wavelength

17. So, What can we do? There are a number of ways to deal with this problem. 1.Compute the entire 3-d image volume. The most straightforward approach, but hugely wasteful in computing resources! The minimum number of ‘vertical planes’ needed is: N n ~ B  2 /  D  The number of volume pixels to be calculated is: N pix ~ 4B 3  4 /  ~ 4 B 3 /D 4 But the number of pixels actually needed is: 4B 2 /D 2 So the fraction of the pixels in the final output map actually used is: D 2 / B. (~ 2% at  = 1 meter in A- configuration!)

18. Deep Cubes! To give an idea of the scale of processing, the table below shows the number of ‘vertical’ planes needed to encompass the VLA’s primary beam. For the A-configuration, each plane is at least 2048 x 2048. For the New Mexico Array, it’s at least 16384 x 16384! And one cube would be needed for each spectral channel, for each polarization! NMAABCDE 400cm2250225682372 90cm5605617621 20cm110114211 6cm4042111 2cm1021111 1.3cm611111

19. 2. Polyhedron Imaging The wasted effort is in computing pixels we don’t need. The polyhedron approach approximates the unit sphere with small flat planes, each of which stays close to the sphere’s surface. For each subimage, the entire dataset must be phase-shifted, and the (u,v,w) recomputed for the new plane. facet

20. Polyhedron Approach, (cont.) How many facets are needed? If we want to minimize distortions, the plane mustn’t depart from the unit sphere by more than the synthesized beam, /B. Simple analysis (see the book) shows the number of facets will be: N f ~ 2 B/D 2 or twice the number needed for 3-D imaging. But the size of each image is much smaller, so the total number of cells computed is much smaller. The extra effort in phase computation and (u,v,w) rotation is more than made up by the reduction in the number of cells computed. This approach is the current standard in AIPS.

21. Polyhedron Imaging Procedure is then: –Determine number of facets, and the size of each. –Generate each facet image, rotating the (u,v,w) and phase- shifting the phase center for each. –Jointly deconvolve the set. The Clark/Cotton/Schwab major/minor cycle system is well suited for this. –Project the finished images onto a 2-d surface. Added benefit of this approach: –As each facet is independently generated, one can imagine a separate antenna-based calibration for each. –Useful if calibration is a function of direction as well as time. –This is needed for meter-wavelength imaging.

22. W-Projection Although the polyhedron approach works well, it is expensive, and there are annoying boundary issues – where the facets overlap. Is it possible to project the data onto a single (u,v) plane, accounting for all the necessary phase shifts? Answer is YES! Tim Cornwell has developed a new algorithm, termed ‘w-projection’, to do this. Available only in CASA (formerly known as AIPS++), this approach permits a single 2-d image and deconvolution, and eliminates the annoying edge effects which accompany re-projection.

23. W-Projection Each visibility, at location (u,v,w) is mapped to the w=0 plane, with a phase shift proportional to the distance. Each visibility is mapped to ALL the points lying within a cone whose full angle is the same as the field of view of the desired map –  /D for a full-field image. Area in the base of the cone is ~4 2 w 2 /D 2 < 4B 2 /D 2. Number of cells on the base which ‘receive’ this visibility is ~ 4w 0 2 B 2 /D 2 < 4B 4 / 2 D 2. w u u 0,w 0 u0u0 u 1,v 1 ~2 /D ~2 w 0 /D

24. W-Projection The phase shift for each visibility onto the w=0 plane is in fact a Fresnel diffraction function. Each 2-d cell receives a value for each observed visibility within an (upward/downwards) cone of full angle  < /D (the antenna’s field of view). In practice, the data are non-uniformly vertically gridded – speeds up the projection. There are a lot of computations, but they are done only once. Spatially-variant self-cal can be accommodated (but hasn’t yet).

25. An Example – without ‘3-D’ Procesesing

26. Example – with 3D processing

27. Conclusion (of sorts) Arrays which measure visibilities within a 3- dimensional (u,v,w) volume, such as the VLA, cannot use a 2-d FFT for wide-field and/or low-frequency imaging. The distortions in 2-d imaging are large, growing quadratically with distance, and linearly with wavelength. In general, a 3-d imaging methodology is necessary. Recent research shows a Fresnel-diffraction projection method is the most efficient, although the older polyhedron method is better known. Undoubtedly, better ways can yet be found.